In the card game Red 7, the deck consists of 49 cards, with cards numbered 1 thr

Question

In the card game Red 7, the deck consists of 49 cards, with cards numbered 1 through 7 in each of the seven colors of the rainbow (red, orange, yellow, green, blue, indigo, violet). All 49 cards are distinct. You are dealt a hand of seven cards.

Answer each question and briefly explain your reasoning. Leave your answer unsimplified in terms of permutations, combinations, exponents, etc.

(a) How many seven-card hands are possible?
(b) How many seven-card hands contain no blue cards?

(c) How many seven-card hands contain all cards of the same color?

(d) How many seven-card hands contain one card of each color?

(e) How many seven-card hands contain at least one red card?

Explanation / Answer

Given number of cards = 49.

Cards of each color = 7.

a.) How many seven-card hand are possible ?

selecting 7 cards out of 49 is

C (49 ,7) -> Here We are Using Combination

so C (49,7) = 85,900,584

b.) How many seven-card hands contain no blue cards ?

So here we will not select blue cards that are 7 .

So remaining cards are 42 and we have to select 7 out of 42

C (42,7) = 26,978,328

c.) How many seven-cards hand contain all cards of same color ?

this is obvious 7 as there are seven colors.

d.) How many seven cards hands contain one card of each color?

7 ^ 7= power(7,7) -> for one card we have seven option each so for seven we have 7^7 ways.

7^7 = 13,841,287,201

e.) How Many seven -cards hands contain at least one red card--

=C(7,1)*C(42,6) + C(7,2)*C(42,5) + C(7,3)*C(42,4) + C(7,4)*C(42,3) + C(7,5)*C(42,2)+ C(7,6)*C(42,1)+C(7,7)

there is all possiblity from one red card to 7 red cards if we choose one red cards then we will choose 6 other card from 42 reamining cards likiewise we can choose upto seven red cards.

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